Mathematical Analysis and Optimization for Economists

Michael J. Panik

• Publisher: CRC Press
• Year: 2021
• Print ISBN: 9780367759018
• eBook ISBN: 9781000408928
• Edition: 1
• Topic: Mathematics
• Subtopic: Operations
• Index: Mathematics

In Mathematical Analysis and Optimization for Economists, the author aims to introduce students of economics to the power and versatility of traditional as well as contemporary methodologies in mathematics and optimization theory; and, illustrates how these techniques can be applied in solving microeconomic problems.

This book combines the areas of intermediate to advanced mathematics, optimization, and microeconomic decision making, and is suitable for advanced undergraduates and first-year graduate students. This text is highly readable, with all concepts fully defined, and contains numerous detailed example problems in both mathematics and microeconomic applications. Each section contains some standard, as well as more thoughtful and challenging, exercises. Solutions can be downloaded from the CRC Press website. All solutions are detailed and complete.

Features

• Contains a whole spectrum of modern applicable mathematical techniques, many of which are not found in other books of this type.
• Comprehensive and contains numerous and detailed example problems in both mathematics and economic analysis.
• Suitable for economists and economics students with only a minimal mathematical background.
• Classroom-tested over the years when the author was actively teaching at the University of Hartford.
• Serves as a beginner text in optimization for applied mathematics students.
• Accompanied by several electronic chapters on linear algebra and matrix theory, nonsmooth optimization, economic efficiency, and distance functions available for free on www.routledge.com/9780367759018.

Chapter 1 Mathematical Foundations 1

1.1 Matrices and Determinants

We start with

Definition 1.1.1: A matrix is an ordered set of elements arranged in a rectangular array of rows and columns.

That is, the matrix A may appear as

$\mathit{A}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right],$

where a_ij represents the element in the ith row and jth column of $\mathit{A},i=1,\dots ,m;j=1,\dots ,n$. Since there are m rows and n columns in A, this matrix is said to be of order (m × n) (“read m by n”). When i, j=1,…,n, the matrix is square and will simply be referred to as an nth order matrix. The matrix A may be written in a more compact fashion as $\mathit{A}=\left[a_{ij}\right],i=1,\dots ,m;j=1,\dots ,n$.

We next have

Definition 1.1.2: The sum of two (m × n) matrices $\mathit{A}=\left[a_{ij}\right],\mathit{B}=\left[b_{ij}\right]$ is the (m × n) matrix $\mathit{C}=\left[c_{ij}\right]$, where $c_{ij}=a_{ij}+b_{ij}$ (we add corresponding elements), i.e.,

$\mathit{A}+\mathit{B}=\mathit{C}\mathrm{or}\left[a_{ij}\right]+\left[b_{ij}\right]=\left[a_{ij}+b_{ij}\right],i=1,\dots ,m;j=1,\dots ,n.$

Definition 1.1.3: The product of a (real) scalar λ and an (m × n) matrix $\mathit{A}=\left[a_{ij}\right]$ is the (m × n)...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author
• Symbols and Abbreviations
• CHAPTER 1 ■ Mathematical Foundations 1
• CHAPTER 2 ■ Mathematical Foundations 2
• CHAPTER 3 ■ Mathematical Foundations 3
• CHAPTER 4 ■ Mathematical Foundations 4
• CHAPTER 5 ■ Global and Local Extrema of Real-Valued Functions
• CHAPTER 6 ■ Global Extrema of Real-Valued Functions
• CHAPTER 7 ■ Local Extrema of Real-Valued Functions
• CHAPTER 8 ■ Convex and Concave Real-Valued Functions
• CHAPTER 9 ■ Generalizations of Convexity and Concavity
• CHAPTER 10 ■ Constrained Extrema: Equality Constraints
• CHAPTER 11 ■ Constrained Extrema: Inequality Constraints
• CHAPTER 12 ■ Constrained Extrema: Mixed Constraints
• CHAPTER 13 ■ Lagrangian Saddle Points and Duality
• CHAPTER 14 ■ Generalized Concave Optimization
• CHAPTER 15 ■ Homogeneous, Homothetic, and Almost Homogeneous Functions
• CHAPTER 16 ■ Envelope Theorems
• CHAPTER 17 ■ The Fixed Point Theorems of Brouwer and Kakutani
• CHAPTER 18 ■ Dynamic Optimization: Optimal Control Modeling
• CHAPTER 19 ■ Comparative Statics Revisited
• REFERENCES
• INDEX